   Chapter 17.1, Problem 26E

Chapter
Section
Textbook Problem

Solve the boundary-value problem, if possible.26. y" + 6y' = 0, y(0) = 1, y(1) = 0

To determine

To solve: The boundary-value problem for differential equation y+6y=0 , y(0)=1 , y(1)=0 .

Explanation

Formula used:

Write the expression for differential equation.

ay+by+cy=0 (1)

Write the expression for auxiliary equation.

ar2+br+c=0 (2)

Write the expression for general solution of ay+by+cy=0 with two distinct real roots r .

y=c1er1x+c2er2x (3)

Here,

r1 and r2 are the root of auxiliary equation.

Consider the differential equation as follows.

y+6y=0 (4)

Compare equation (1) and (4).

a=1b=6c=0

Find the auxiliary equation.

Substitute 1 for a , 6 for b and 0 for c in equation (2),

(1)r2+(6)r+(0)=0r2+6r=0r(r+6)=0

Simplify equation as follows.

r=0r+6=0r=6

Consider the value of r1 and r2 as follows.

r1=0r2=6

Find the general solution of y+6y=0 using equation (3).

Substitute 0 for r1 and 6 for r2 in equation (3),

y=c1e(0)x+c2e(6)x

y=c1+c2e6x (5)

Modify equation (5) as follows.

y(x)=c1+c2e6x (6)

Find the value of y(0) .

Substitute 0 for x in equation (6),

y(0)=c1+c2e6(0)=c1+c2

Substitute 1 for y(0) ,

1=c1+c2 (7)

Find the value of y(1)

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